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 A184837 a(n) = n + floor(n*t) + floor(n*t^2) + floor(n/t) + floor(n/t^2), where t is the pentanacci constant. 5

%I #6 Mar 30 2012 18:37:25

%S 5,13,20,29,36,44,51,59,66,74,81,90,97,105,111,120,127,135,142,151,

%T 158,166,172,181,188,196,203,212,219,225,233,241,248,256,264,272,279,

%U 286,294,302,309,317,325,333,339,347,355,363,370,378,386,393,400,408,416,424,431,440,447,453,461,469,477,484,492,500,507,514,522,530,538,545,553,561,568,575,583,591,599,606,614,621,629,636,644,652,660,667,674,681,689,696,705,712,720,727,735,742,750,757,766,773,781,787,796,803

%N a(n) = n + floor(n*t) + floor(n*t^2) + floor(n/t) + floor(n/t^2), where t is the pentanacci constant.

%C This is one of five sequences that partition the positive integers.

%C Given t is the pentanacci constant, then the following sequences are disjoint:

%C . A184835(n) = n + [n/t] + [n/t^2] + [n/t^3] + [n/t^4],

%C . A184836(n) = n + [n*t] + [n/t] + [n/t^2] + [n/t^3],

%C . A184837(n) = n + [n*t] + [n*t^2] + [n/t] + [n/t^2],

%C . A184838(n) = n + [n*t] + [n*t^2] + [n*t^3] + [n/t],

%C . A184839(n) = n + [n*t] + [n*t^2] + [n*t^3] + [n*t^4], where []=floor.

%C This is a special case of Clark Kimberling's results given in A184812.

%F Limit a(n)/n = t^3 = 7.5982964914823797216620775...

%F a(n) = n + floor(n*p/r) + floor(n*q/r) + floor(n*s/r) + floor(n*u/r), where p=t, q=t^2, r=t^3, s=t^4, u=t^5, and t is the pentanacci constant.

%e Given t = pentanacci constant, then t^3 = 1 + t + t^2 + 1/t + 1/t^2,

%e t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

%o (PARI) {a(n)=local(t=real(polroots(1+x+x^2+x^3+x^4-x^5)[1])); n+floor(n*t)+floor(n*t^2)+floor(n/t)+floor(n/t^2)}

%Y Cf. A184835, A184836, A184838, A184839; A184812, A103814.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jan 23 2011

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